Bounds, Heuristics, and Prophet Inequalities for Assortment Optimization PDF Download
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Author: Guillermo Gallego
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Languages : en
Pages : 0
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We address two important concerns faced by assortment managers, namely constrained assortment optimization and assortment personalization. We contribute to addressing these concerns by developing bounds and heuristics based on auxiliary multinomial logit (MNL) models. More precisely, we first provide easily computable upper and lower bounds for the unconstrained assortment optimization problem (TAOP) for every regular choice model and then extend the bounds to important versions of the constrained problem. We next provide an upper bound on the expected revenue of a clairvoyant firm that offers to each consumer the most profitable product that she is willing to buy. We then use the upper bound to assess the maximum benefits of personalization relative to a firm that does not personalize assortments. The standard prophet inequality is then used to show that the ratio is at most 2 for discrete choice models with { em independent} value gaps. For random utility models with dependent value gaps the ratio can be as large as the number of products. We find sufficient conditions to show that the prophet inequality holds for the $ alpha$-shaken multinomial logit ($ alpha$-MNL), a generalization of the MNL introduced here, that has the MNL and the generalized attraction model (GAM) as special cases. The prophet inequality also holds for the some versions of the Nested Logit model. For the latent-class MNL, the ratio is at most 1.5 when the coefficient of variation of the utilities goes to infinity. We show that consumers do not necessarily suffer under a clairvoyant firm and in fact their surplus may improve. On the other hand, when the clairvoyant firm has pricing power it can extract all of the consumers' surplus. We show that for the MNL model the clairvoyant can make up to $e$ times more than its non-clairvoyant counterpart.
Author: Guillermo Gallego
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Languages : en
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Book Description
We address two important concerns faced by assortment managers, namely constrained assortment optimization and assortment personalization. We contribute to addressing these concerns by developing bounds and heuristics based on auxiliary multinomial logit (MNL) models. More precisely, we first provide easily computable upper and lower bounds for the unconstrained assortment optimization problem (TAOP) for every regular choice model and then extend the bounds to important versions of the constrained problem. We next provide an upper bound on the expected revenue of a clairvoyant firm that offers to each consumer the most profitable product that she is willing to buy. We then use the upper bound to assess the maximum benefits of personalization relative to a firm that does not personalize assortments. The standard prophet inequality is then used to show that the ratio is at most 2 for discrete choice models with { em independent} value gaps. For random utility models with dependent value gaps the ratio can be as large as the number of products. We find sufficient conditions to show that the prophet inequality holds for the $ alpha$-shaken multinomial logit ($ alpha$-MNL), a generalization of the MNL introduced here, that has the MNL and the generalized attraction model (GAM) as special cases. The prophet inequality also holds for the some versions of the Nested Logit model. For the latent-class MNL, the ratio is at most 1.5 when the coefficient of variation of the utilities goes to infinity. We show that consumers do not necessarily suffer under a clairvoyant firm and in fact their surplus may improve. On the other hand, when the clairvoyant firm has pricing power it can extract all of the consumers' surplus. We show that for the MNL model the clairvoyant can make up to $e$ times more than its non-clairvoyant counterpart.
Author: Sumit Kunnumkal
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Languages : en
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We consider the assortment optimization problem under the nested logit model and obtain new bounds on the gap between the optimal expected revenue and an upper bound based on a certain continuous relaxation of the assortment problem. Our bounds can be tighter than the existing bounds in the literature and provide more insight into the key drivers of tractability for the assortment optimization problem under the nested logit model. Moreover, our bounds scale with the nest dissimilarity parameters and we recover the well-known tractability results for the assortment optimization problem under the multinomial logit model when all the nest dissimilarity parameters are equal to one. We extend our results to the cardinality constrained assortment problem where there are constraints that limit the number of products that can be offered within each nest.
Author: Clark Pixton
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Languages : en
Pages : 29
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We study the static assortment optimization problem under weakly rational choice models, i.e. models in which adding a product to an assortment does not increase the probability of purchasing a product already in that assortment. This setting applies to most choice models studied and used in practice, such as the multinomial logit and random parameters logit models. We give a mixed-integer linear optimization formulation with an exponential number of constraints, and present two branch-and-bound algorithms for solving this optimization problem. The formulation and algorithms require only black-box access to purchase probabilities, and thus provide exact solution methods for a general class of discrete choice models, in particular those models without closed-form choice probabilities. We show that one of our algorithms is a PTAS for assortment optimization under weakly rational choice when the no-purchase probability is small, and give an approximation guarantee for the other algorithm which depends only on the prices of the products. Finally, we test the performance of our algorithms with heuristic stopping criteria, motivated by the fact that they discover the optimal solution very quickly.
Author: Asrar Ahmed
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Languages : en
Pages : 0
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We consider the problem faced by an online service platform that matches suppliers with consumers. Unlike traditional matching models, which treat them as passive participants, we allow both sides of the market to exercise their choices. To model this setting, we introduce a two-sided assortment optimization model wherein each participant's choice is modeled using a multinomial logit choice function, and the platform's objective is to maximize its expected revenue. We first show that the problem is NP-hard even when the number of suppliers is limited to two and provide a mixed-integer linear programming formulation. Next, we discuss two simple greedy heuristics and argue that these can lead to arbitrarily bad solutions. We then develop relaxations that provide upper and lower bounds and investigate the tightness of these relaxations by obtaining parametric approximation guarantees. Finally, we present numerical results on synthetic data demonstrating the practical utility of these relaxations.
Author: Guillermo Gallego
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Languages : en
Pages : 0
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We study the assortment optimization problem and show that local optima are global optima for all discrete choice models that can be represented by the Markov Chain model. We develop a forward greedy heuristic that finds an optimal assortment for the Markov Chain model and runs in $O(n^2)$ iterations. The heuristic has performance bound $1/n$ for any regular choice model which is best possible among polynomial heuristics. We also propose a backward greedy heuristic that is optimal for Markov chain model and requires fewer iterations. Numerical results show that our heuristics performs significantly better than the estimate then optimize method and the revenue-ordered assortment heuristic when the ground truth is a latent class multinomial logit choice model. Based on the greedy heuristics, we develop a learning algorithm that enjoys asymptotic optimal regret for the Markov chain choice model and avoids parameter estimations, focusing instead on binary comparisons of revenues.
Author: Laurent Alfandari
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Languages : en
Pages : 39
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We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a Nested Logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. For the NP-hard cases, we provide a general exact method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. Contrary to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed, and hence our approach can solve the most general problem setting. We show that the parameterized subproblem of the fractional programming scheme, which is a binary highly non-linear optimization problem, is decomposable by nests, which is a main advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we identify those products that are undoubtedly beneficial to offer, or not, which can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with up to 5,000 products per nest. The most challenging instances for our approach are those in which the dissimilarity parameters of nests can be either less or greater than one.
Author: Ali Aouad
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Languages : en
Pages : 20
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The main contribution of this paper is to provide best-possible approximability bounds for assortment planning under a general choice model, where customer choices are modeled through an arbitrary distribution over ranked lists of their preferred products, subsuming most random utility choice models of interest. From a technical perspective, we show how to relate this optimization problem to the computational task of detecting large independent sets in graphs, allowing us to argue that general ranking preferences are extremely hard to approximate with respect to various problem parameters. These findings are complemented by a number of approximation algorithms that attain essentially best-possible factors, proving that our hardness results are tight up to lower-order terms. Surprisingly, our results imply that a simple and widely studied policy, known as revenue-ordered assortments, achieves the best possible performance guarantee with respect to the price parameters.
Author: Srikanth Jagabathula
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Languages : en
Pages : 51
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We consider the key operational problem of optimizing the mix of offered products to maximize revenues when product prices are exogenously set and product demand follows a general discrete choice model. The key challenge in making this decision is the computational difficulty of finding the best subset, which often requires exhaustive search. Existing approaches address the challenge by either deriving efficient algorithms for specific parametric structures or studying the performance of general-purpose heuristics. The former approach results in algorithms that lack portability to other structures; whereas the latter approach has resulted in algorithms with poor performance. We study a portable and easy-to-implement local search heuristic. We show that it efficiently finds the global optimum for the multinomial logit (MNL) model and derive performance guarantees for general choice structures. Empirically, it is better than prevailing heuristics when no efficient algorithms exist, and it is within 0.02% of optimality otherwise.
Author: Tien Mai
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Languages : en
Pages : 37
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This work concerns the assortment optimization problem that refers to selecting a subset of items that maximizes the expected revenue in the presence of the substitution behavior of consumers specified by a parametric choice model. The key challenge lies in the computational difficulty of finding the best subset solution, which often requires exhaustive search. The literature on constrained assortment optimization lacks a practically efficient method which that is general to deal with different types of parametric choice models (e.g., the multinomial logit, mixed logit or general multivariate extreme value models). In this paper, we propose a new approach that allows to address this issue. The idea is that, under a general parametric choice model, we formulate the problem into a binary nonlinear programming model, and use an iterative algorithm to find a binary solution. At each iteration, we propose a way to approximate the objective (expected revenue) by a linear function, and a polynomial-time algorithm to find a candidate solution using this approximate function. We also develop a greedy local search algorithm to further improve the solutions. We test our algorithm on instances of different sizes under various parametric choice model structures and show that our algorithm dominates existing exact and heuristic approaches in the literature, in terms of solution quality and computing cost.
Author: Mohammed Ali Aouad
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Languages : en
Pages : 256
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Finding optimal product offerings is a fundamental operational issue in modern retailing, exemplified by the development of recommendation systems and decision support tools. The challenge is that designing an accurate predictive choice model generally comes at the detriment of efficient algorithms, which can prescribe near-optimal decisions. This thesis attempts to resolve this disconnect in the context of assortment and inventory optimization, through theoretical and empirical investigation. First, we tightly characterize the complexity of general nonparametric assortment optimization problems. We reveal connections to maximum independent set and combinatorial pricing problems, allowing to derive strong inapproximability bounds. We devise simple algorithms that achieve essentially best-possible factors with respect to the price ratio, size of customers' consideration sets, etc. Second, we develop a novel tractable approach to choice modeling, in the vein of nonparametric models, by leveraging documented assumptions on the customers' consider-then-choose behavior. We show that the assortment optimization problem can be cast as a dynamic program, that exploits the properties of a bi-partite graph representation to perform a state space collapse. Surprisingly, this exact algorithm is provably and practically efficient under common consider-then-choose assumptions. On the estimation front, we show that a critical step of standard nonparametric estimation methods (rank aggregation) can be solved in polynomial time in settings of interest, contrary to general nonparametric models. Predictive experiments on a large purchase panel dataset show significant improvements against common benchmarks. Third, we turn our attention to joint assortment optimization and inventory management problems under dynamic customer choice substitution. Prior to our work, little was known about these optimization models, which are intractable using modern discrete optimization solvers. Using probabilistic analysis, we unravel hidden structural properties, such as weak notions of submodularity. Building on these findings, we develop efficient and yet conceptually-simple approximation algorithms for common parametric and nonparametric choice models. Among notable results, we provide best-possible approximations under general nonparametric choice models (up to lower-order terms), and develop the first constant-factor approximation under the popular Multinomial Logit model. In synthetic experiments vis-a-vis existing heuristics, our approach is an order of magnitude faster in several cases and increases revenue by 6% to 16%.
